Generalized permutation matrix: Difference between revisions

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Revision as of 17:32, 29 July 2013

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{{unreferenced|date=August 2009}}

In [[mathematics]], a '''generalized permutation matrix''' (or '''monomial matrix''') is a [[matrix (mathematics)|matrix]] with the same nonzero pattern as a [[permutation matrix]], i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero entry can be any nonzero value. An example of a generalized permutation matrix is

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{{main|Monomial representation}}

Monomial matrices occur in [[representation theory]] in the context of [[monomial representation]]s. A monomial representation of a group ''G'' is a linear representation ''ρ'' : ''G'' → GL(''n'', ''F'') of ''G'' (here ''F'' is the defining field of the representation) such that the image ''ρ''(''G'') is a subgroup of the group of monomial matrices.

==References==

* {{cite book | last=Joyner | first=David | title=Adventures in group theory. Rubik's cube, Merlin's machine, and other mathematical toys | edition=2nd updated and revised | ___location=Baltimore, MD | publisher=Johns Hopkins University Press | year=2008 | isbn=978-0-8018-9012-3 | zbl=1221.00013 }}

[[Category:Matrices]]